Question

Step-1: Drawing a Quadrilateral

o Draw a horizontal line AB of 5 cm length
o Draw a ray ‘Ax’ from point A at an angle of
110° (using protractor)
o Draw a second ray ‘By’ from point B at an
angle of 70° (using protractor)
o Cut the both these rays with arcs of 4 cm
radius from points A & B respectively
o Mark the left side intersection point as D and
the right side intersection point as C
o Join the points C and D with scale
Now, we have a quadrilateral (i.e. polygon with 4
sides)
• Step-2: Identifying the type of quadrilateral
o Join A to C (what do we call the line segment
AC?)
o Measure angles ∠ACD and ∠CAB (How do
they compare with each other?)
o Measure angles ∠CAD and ∠ACB (How do
they compare with each other?)
∴Line ABCD and line ADBC
Quadrilateral in which each pair of opposite sides are
parallel is known as PARALLELOGRAM
• Step-3: Divide the parallelogram along the
diagonal
o Diagonal AC has divided the parallelogram
ABCD into a set of 2 triangles i.e. ∆ABC and
∆ADC
o Colour both triangles in two different shades
o Measure their sides and angles. Tabulate them
side by side in your note book
o Calculate areas both triangles using Heron’s
formula and write down in your note book
Now, you might have observed that areas of ∆ABC
and ∆ADC are same. If so, are these two ∆s congruent
to each other? Let’s test it!
• Step-4: Properties of a Parallelogram
Concept congruency of triangles
1. Four criteria Viz. SSS, SAS, ASA & AAS are
used to determine if two ∆s are congruent
2. If ∆s are found to be congruent, then their
areas will be equal
3. Here, let us use the criteria SSS to check the
congruency of these two ∆s, ∆ABC and
∆ADC.
4. Length measurements from the note book
reveal that length of AB and CD are 5cm.

5. Similarly, length of AD and BC are 4 cm.
6. AC (the diagonal) is common between both
∆s
∴ Three sides of ∆ABC are same as that of ∆ADC
It implies that both ∆ABC and ∆ADC are congruent
to each other.
It also confirms that the areas calculated using
Heron’s formula earlier proves and confirms that the
two ∆s ∆ABC and ∆ADC are congruent to each other.
Property-1: So, if two congruent triangles are
formed by the diagonal of a quadrilateral, such a
quadrilateral will have to be a parallelogram
o It is also seen from the measurement table
(refer point 4 of step-4) that both AB & CD are
equal in length (5 cm) and AD & BC are aslo
equal in length (4 cm)
o Step-2 has proved that line ABCD and line
ADBC
Property-2: So, in a quadrilateral, if each pair of
opposite sides are parallel and equal, then it is said
to be a parallelogram
o Step-2 has shown that angles ∠ACD = ∠CAB
and angles ∠CAD = ∠ACB
∴∠ACD + ∠ACB = ∠CAB + ∠CAD
o Draw a diagonal from B to D, and tabulate the
measurements for its angles
o It will show you that ∠ADB + ∠BDC = ∠ABD
+ ∠DBC
Property-3: In a quadrilateral, if each pair of
opposite angles are equal, then it is said to be a
parallelogram
o Mark the point of intersection of both
diagonals i.e. AC and BD as O.
o Tabulate the following measurements AO, OC
and BO, OD and also AC and BD
o Now you can see that AO = OC and BO = OD
∴ it can be concluded that diagonals AC and BD
bisect each other
However, AC≠BD
Property-4: In a quadrilateral, if both diagonals
bisect each other and are of different lengths, then it
is said to be a parallelogram

Answer 1

Answer:

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